Optimal Transport Methods in Operations Research and Statistics - PowerPoint PPT Presentation

Proof Techniques Suppose X and Y compact � � sup inf X ×Y c ( x , y ) π ( dx , dy ) α , β π ≥ 0 , � � − X ×Y α ( x ) π ( dx , dy ) + X α ( x ) µ ( dx ) � � − X ×Y β ( y ) π ( dx , dy ) + Y β ( y ) v ( dy ) } Swap sup and inf using Sion’s min-max theorem by a compactness argument and conclude. Significant amount of work needed to extend to general Polish spaces and construct the dual optimizers (primal a bit easier). Blanchet (Columbia U. and Stanford U.) 12 / 60

Optimal Transport Applications Optimal Transport has gained popularity in many areas including: image analysis, economics, statistics, machine learning... The rest of the talk mostly concerns applications to OR and Statistics but we’ll briefly touch upon others, including economics... Blanchet (Columbia U. and Stanford U.) 13 / 60

Illustration of Optimal Transport in Image Analysis Santambrogio (2010)’s illustration Blanchet (Columbia U. and Stanford U.) 14 / 60

Application of Optimal Transport in Economics Economic Interpretations (see e.g. A. Galichon’s 2016 textbook & McCaan 2013 notes). Blanchet (Columbia U. and Stanford U.) 15 / 60

Applications in Labor Markets Worker with skill x & company with technology y have surplus Ψ ( x , y ) . Blanchet (Columbia U. and Stanford U.) 16 / 60

Applications in Labor Markets Worker with skill x & company with technology y have surplus Ψ ( x , y ) . The population of workers is given by µ ( x ) . Blanchet (Columbia U. and Stanford U.) 16 / 60

Applications in Labor Markets Worker with skill x & company with technology y have surplus Ψ ( x , y ) . The population of workers is given by µ ( x ) . The population of companies is given by v ( y ) . Blanchet (Columbia U. and Stanford U.) 16 / 60

Applications in Labor Markets Worker with skill x & company with technology y have surplus Ψ ( x , y ) . The population of workers is given by µ ( x ) . The population of companies is given by v ( y ) . The salary of worker x is α ( x ) & cost of technology y is β ( y ) α ( x ) + β ( y ) ≥ Ψ ( x , y ) . Blanchet (Columbia U. and Stanford U.) 16 / 60

Applications in Labor Markets Worker with skill x & company with technology y have surplus Ψ ( x , y ) . The population of workers is given by µ ( x ) . The population of companies is given by v ( y ) . The salary of worker x is α ( x ) & cost of technology y is β ( y ) α ( x ) + β ( y ) ≥ Ψ ( x , y ) . Companies want to minimize total production cost � � α ( x ) µ ( x ) dx + β ( y ) v ( y ) dy Blanchet (Columbia U. and Stanford U.) 16 / 60

Applications in Labor Markets Letting a central planner organize the Labor market Blanchet (Columbia U. and Stanford U.) 17 / 60

Applications in Labor Markets Letting a central planner organize the Labor market The planner wishes to maximize total surplus � Ψ ( x , y ) π ( dx , dy ) Blanchet (Columbia U. and Stanford U.) 17 / 60

Applications in Labor Markets Letting a central planner organize the Labor market The planner wishes to maximize total surplus � Ψ ( x , y ) π ( dx , dy ) Over assignments π ( · ) which satisfy market clearing � � Y π ( dx , dy ) = µ ( dx ) , X π ( dx , dy ) = v ( dy ) . Blanchet (Columbia U. and Stanford U.) 17 / 60

Solving for Optimal Transport Coupling Suppose that Ψ ( x , y ) = xy , µ ( x ) = I ( x ∈ [ 0 , 1 ]) , v ( y ) = e − y I ( y > 0 ) . Blanchet (Columbia U. and Stanford U.) 18 / 60

Solving for Optimal Transport Coupling Suppose that Ψ ( x , y ) = xy , µ ( x ) = I ( x ∈ [ 0 , 1 ]) , v ( y ) = e − y I ( y > 0 ) . i } n i } n Solve primal by sampling: Let { X n i = 1 and { Y n i = 1 both i.i.d. from µ and v , respectively. n n � � F µ n ( x ) = 1 i ≤ x ) , F v n ( y ) = 1 I ( X n Y n ∑ ∑ I j ≤ y n n i = 1 j = 1 Blanchet (Columbia U. and Stanford U.) 18 / 60

Solving for Optimal Transport Coupling Suppose that Ψ ( x , y ) = xy , µ ( x ) = I ( x ∈ [ 0 , 1 ]) , v ( y ) = e − y I ( y > 0 ) . i } n i } n Solve primal by sampling: Let { X n i = 1 and { Y n i = 1 both i.i.d. from µ and v , respectively. n n � � F µ n ( x ) = 1 i ≤ x ) , F v n ( y ) = 1 I ( X n Y n ∑ ∑ I j ≤ y n n i = 1 j = 1 Consider � � � � x n i , y n x n i , y n j ) ≥ 0 ∑ max Ψ π j j π ( x n i , x n i , j � � = 1 � � = 1 n ∀ x i , ∑ x n i , y n x n i , y n ∑ π π n ∀ y j . j j j i Blanchet (Columbia U. and Stanford U.) 18 / 60

Solving for Optimal Transport Coupling Suppose that Ψ ( x , y ) = xy , µ ( x ) = I ( x ∈ [ 0 , 1 ]) , v ( y ) = e − y I ( y > 0 ) . i } n i } n Solve primal by sampling: Let { X n i = 1 and { Y n i = 1 both i.i.d. from µ and v , respectively. n n � � F µ n ( x ) = 1 i ≤ x ) , F v n ( y ) = 1 I ( X n Y n ∑ ∑ I j ≤ y n n i = 1 j = 1 Consider � � � � x n i , y n x n i , y n j ) ≥ 0 ∑ max Ψ π j j π ( x n i , x n i , j � � = 1 � � = 1 n ∀ x i , ∑ x n i , y n x n i , y n ∑ π π n ∀ y j . j j j i Clearly, simply sort and match is the solution! Blanchet (Columbia U. and Stanford U.) 18 / 60

Solving for Optimal Transport Coupling � � Think of Y n 1 − U n for U n j = − log j s i.i.d. uniform ( 0 , 1 ) . j Blanchet (Columbia U. and Stanford U.) 19 / 60

Solving for Optimal Transport Coupling � � Think of Y n 1 − U n for U n j = − log j s i.i.d. uniform ( 0 , 1 ) . j The j -th order statistic X n ( j ) is matched to Y n ( j ) . Blanchet (Columbia U. and Stanford U.) 19 / 60

Solving for Optimal Transport Coupling � � Think of Y n 1 − U n for U n j = − log j s i.i.d. uniform ( 0 , 1 ) . j The j -th order statistic X n ( j ) is matched to Y n ( j ) . As n → ∞ , X n ( nt ) → t , so Y n ( nt ) → − log ( 1 − t ) . Blanchet (Columbia U. and Stanford U.) 19 / 60

Solving for Optimal Transport Coupling � � Think of Y n 1 − U n for U n j = − log j s i.i.d. uniform ( 0 , 1 ) . j The j -th order statistic X n ( j ) is matched to Y n ( j ) . As n → ∞ , X n ( nt ) → t , so Y n ( nt ) → − log ( 1 − t ) . Thus, the optimal coupling as n → ∞ is X = U and Y = − log ( 1 − U ) (comonotonic coupling). Blanchet (Columbia U. and Stanford U.) 19 / 60

Identities for Wasserstein Distances Comonotonic coupling is the solution if ∂ 2 x , y Ψ ( x , y ) ≥ 0 - supermodularity. Blanchet (Columbia U. and Stanford U.) 20 / 60

Identities for Wasserstein Distances Comonotonic coupling is the solution if ∂ 2 x , y Ψ ( x , y ) ≥ 0 - supermodularity. Of for costs c ( x , y ) = − Ψ ( x , y ) if ∂ 2 x , y c ( x , y ) ≤ 0 (submodularity). Blanchet (Columbia U. and Stanford U.) 20 / 60

Identities for Wasserstein Distances Comonotonic coupling is the solution if ∂ 2 x , y Ψ ( x , y ) ≥ 0 - supermodularity. Of for costs c ( x , y ) = − Ψ ( x , y ) if ∂ 2 x , y c ( x , y ) ≤ 0 (submodularity). Corollary: Suppose c ( x , y ) = | x − y | then X = F − 1 ( U ) and µ Y = F − 1 ( U ) thus v � 1 � � � � = � � � F − 1 ( u ) − F − 1 D c F µ , F v ( u ) � du . µ v 0 Blanchet (Columbia U. and Stanford U.) 20 / 60

Identities for Wasserstein Distances Comonotonic coupling is the solution if ∂ 2 x , y Ψ ( x , y ) ≥ 0 - supermodularity. Of for costs c ( x , y ) = − Ψ ( x , y ) if ∂ 2 x , y c ( x , y ) ≤ 0 (submodularity). Corollary: Suppose c ( x , y ) = | x − y | then X = F − 1 ( U ) and µ Y = F − 1 ( U ) thus v � 1 � � � � = � � � F − 1 ( u ) − F − 1 D c F µ , F v ( u ) � du . µ v 0 Similar identities are common for Wasserstein distances... Blanchet (Columbia U. and Stanford U.) 20 / 60

Interesting Insight on Salary Effects In equilibrium, by the envelope theorem β ∗ ( y ) = d x [ Ψ ( x , y ) − λ ∗ ( x )] = ∂ ˙ ∂ y Ψ ( x y , y ) = x y . dy sup Blanchet (Columbia U. and Stanford U.) 21 / 60

Interesting Insight on Salary Effects In equilibrium, by the envelope theorem β ∗ ( y ) = d x [ Ψ ( x , y ) − λ ∗ ( x )] = ∂ ˙ ∂ y Ψ ( x y , y ) = x y . dy sup We also know that y = − log ( 1 − x ) , or x = 1 − exp ( − y ) β ∗ ( y ) y + exp ( − y ) − 1 + β ∗ ( 0 ) . = α ∗ ( x ) + β ∗ ( − log ( 1 − x )) = xy . Blanchet (Columbia U. and Stanford U.) 21 / 60

Interesting Insight on Salary Effects In equilibrium, by the envelope theorem β ∗ ( y ) = d x [ Ψ ( x , y ) − λ ∗ ( x )] = ∂ ˙ ∂ y Ψ ( x y , y ) = x y . dy sup We also know that y = − log ( 1 − x ) , or x = 1 − exp ( − y ) β ∗ ( y ) y + exp ( − y ) − 1 + β ∗ ( 0 ) . = α ∗ ( x ) + β ∗ ( − log ( 1 − x )) = xy . What if Ψ ( x , y ) → Ψ ( x , y ) + f ( x ) ? (i.e. productivity grows). Blanchet (Columbia U. and Stanford U.) 21 / 60

Interesting Insight on Salary Effects In equilibrium, by the envelope theorem β ∗ ( y ) = d x [ Ψ ( x , y ) − λ ∗ ( x )] = ∂ ˙ ∂ y Ψ ( x y , y ) = x y . dy sup We also know that y = − log ( 1 − x ) , or x = 1 − exp ( − y ) β ∗ ( y ) y + exp ( − y ) − 1 + β ∗ ( 0 ) . = α ∗ ( x ) + β ∗ ( − log ( 1 − x )) = xy . What if Ψ ( x , y ) → Ψ ( x , y ) + f ( x ) ? (i.e. productivity grows). Answer: salaries grows if f ( · ) is increasing. Blanchet (Columbia U. and Stanford U.) 21 / 60

Applications of Optimal Transport in Stochastic OR Application of Optimal Transport in Stochastic OR Blanchet and Murthy (2016) https://arxiv.org/abs/1604.01446. Insight: Diffusion approximations and optimal transport Blanchet (Columbia U. and Stanford U.) 22 / 60

A Distributionally Robust Performance Analysis In Stochastic OR we are often interested in evaluating E P true ( f ( X )) for a complex model P true Blanchet (Columbia U. and Stanford U.) 23 / 60

A Distributionally Robust Performance Analysis In Stochastic OR we are often interested in evaluating E P true ( f ( X )) for a complex model P true Moreover, we wish to control / optimize it min θ E P true ( h ( X , θ )) . Blanchet (Columbia U. and Stanford U.) 23 / 60

A Distributionally Robust Performance Analysis In Stochastic OR we are often interested in evaluating E P true ( f ( X )) for a complex model P true Moreover, we wish to control / optimize it min θ E P true ( h ( X , θ )) . Model P true might be unknown or too difficult to work with. Blanchet (Columbia U. and Stanford U.) 23 / 60

A Distributionally Robust Performance Analysis In Stochastic OR we are often interested in evaluating E P true ( f ( X )) for a complex model P true Moreover, we wish to control / optimize it min θ E P true ( h ( X , θ )) . Model P true might be unknown or too difficult to work with. So, we introduce a proxy P 0 which provides a good trade-off between tractability and model fidelity (e.g. Brownian motion for heavy-traffic approximations). Blanchet (Columbia U. and Stanford U.) 23 / 60

A Distributionally Robust Performance Analysis For f ( · ) upper semicontinuous with E P 0 | f ( X ) | < ∞ sup E P ( f ( Y )) D c ( P , P 0 ) ≤ δ , X takes values on a Polish space and c ( · ) is lower semi-continuous. Blanchet (Columbia U. and Stanford U.) 24 / 60

A Distributionally Robust Performance Analysis For f ( · ) upper semicontinuous with E P 0 | f ( X ) | < ∞ sup E P ( f ( Y )) D c ( P , P 0 ) ≤ δ , X takes values on a Polish space and c ( · ) is lower semi-continuous. Also an infinite dimensional linear program � sup X ×Y f ( y ) π ( dx , dy ) � s . t . X ×Y c ( x , y ) π ( dx , dy ) ≤ δ � Y π ( dx , dy ) = P 0 ( dx ) . Blanchet (Columbia U. and Stanford U.) 24 / 60

A Distributionally Robust Performance Analysis Formal duality: � � � = λδ + α ( x ) P 0 ( dx ) Dual inf λ ≥ 0 , α λ c ( x , y ) + α ( x ) ≥ f ( y ) . Blanchet (Columbia U. and Stanford U.) 25 / 60

A Distributionally Robust Performance Analysis Formal duality: � � � = λδ + α ( x ) P 0 ( dx ) Dual inf λ ≥ 0 , α λ c ( x , y ) + α ( x ) ≥ f ( y ) . B. & Murthy (2016) - No duality gap : � � �� Dual = inf λδ + E 0 sup y { f ( y ) − λ c ( X , y ) } . λ ≥ 0 Blanchet (Columbia U. and Stanford U.) 25 / 60

A Distributionally Robust Performance Analysis Formal duality: � � � = λδ + α ( x ) P 0 ( dx ) Dual inf λ ≥ 0 , α λ c ( x , y ) + α ( x ) ≥ f ( y ) . B. & Murthy (2016) - No duality gap : � � �� Dual = inf λδ + E 0 sup y { f ( y ) − λ c ( X , y ) } . λ ≥ 0 We refer to this as RoPA Duality in this talk. Blanchet (Columbia U. and Stanford U.) 25 / 60

A Distributionally Robust Performance Analysis Formal duality: � � � = λδ + α ( x ) P 0 ( dx ) Dual inf λ ≥ 0 , α λ c ( x , y ) + α ( x ) ≥ f ( y ) . B. & Murthy (2016) - No duality gap : � � �� Dual = inf λδ + E 0 sup y { f ( y ) − λ c ( X , y ) } . λ ≥ 0 We refer to this as RoPA Duality in this talk. Let us consider the important case f ( y ) = I ( y ∈ A ) & c ( x , x ) = 0. Blanchet (Columbia U. and Stanford U.) 25 / 60

A Distributionally Robust Performance Analysis So, if f ( y ) = I ( y ∈ A ) and c A ( X ) = inf { y ∈ A : c ( x , y ) } , then � λδ + E 0 ( 1 − λ c A ( X )) + � Dual = inf = P 0 ( c A ( X ) ≤ 1 / λ ∗ ) . λ ≥ 0 Blanchet (Columbia U. and Stanford U.) 26 / 60

A Distributionally Robust Performance Analysis So, if f ( y ) = I ( y ∈ A ) and c A ( X ) = inf { y ∈ A : c ( x , y ) } , then � λδ + E 0 ( 1 − λ c A ( X )) + � Dual = inf = P 0 ( c A ( X ) ≤ 1 / λ ∗ ) . λ ≥ 0 If c A ( X ) is continuous under P 0 & E 0 ( c A ( X )) ≥ δ , then δ = E 0 [ c A ( X ) I ( c A ( X ) ≤ 1 / λ ∗ )] . Blanchet (Columbia U. and Stanford U.) 26 / 60

Example: Model Uncertainty in Bankruptcy Calculations R ( t ) = the reserve (perhaps multiple lines) at time t . Blanchet (Columbia U. and Stanford U.) 27 / 60

Example: Model Uncertainty in Bankruptcy Calculations R ( t ) = the reserve (perhaps multiple lines) at time t . Bankruptcy probability (in finite time horizon T ) u T = P true ( R ( t ) ∈ B for some t ∈ [ 0 , T ]) . Blanchet (Columbia U. and Stanford U.) 27 / 60

Example: Model Uncertainty in Bankruptcy Calculations R ( t ) = the reserve (perhaps multiple lines) at time t . Bankruptcy probability (in finite time horizon T ) u T = P true ( R ( t ) ∈ B for some t ∈ [ 0 , T ]) . B is a set which models bankruptcy. Blanchet (Columbia U. and Stanford U.) 27 / 60

Example: Model Uncertainty in Bankruptcy Calculations R ( t ) = the reserve (perhaps multiple lines) at time t . Bankruptcy probability (in finite time horizon T ) u T = P true ( R ( t ) ∈ B for some t ∈ [ 0 , T ]) . B is a set which models bankruptcy. Problem: Model ( P true ) may be complex, intractable or simply unknown... Blanchet (Columbia U. and Stanford U.) 27 / 60

A Distributionally Robust Risk Analysis Formulation Our solution: Estimate u T by solving sup P true ( R ( t ) ∈ B for some t ∈ [ 0 , T ]) , D c ( P 0 , P ) ≤ δ where P 0 is a suitable model. Blanchet (Columbia U. and Stanford U.) 28 / 60

A Distributionally Robust Risk Analysis Formulation Our solution: Estimate u T by solving sup P true ( R ( t ) ∈ B for some t ∈ [ 0 , T ]) , D c ( P 0 , P ) ≤ δ where P 0 is a suitable model. P 0 = proxy for P true . Blanchet (Columbia U. and Stanford U.) 28 / 60

A Distributionally Robust Risk Analysis Formulation Our solution: Estimate u T by solving sup P true ( R ( t ) ∈ B for some t ∈ [ 0 , T ]) , D c ( P 0 , P ) ≤ δ where P 0 is a suitable model. P 0 = proxy for P true . P 0 right trade-off between fidelity and tractability. Blanchet (Columbia U. and Stanford U.) 28 / 60

A Distributionally Robust Risk Analysis Formulation Our solution: Estimate u T by solving sup P true ( R ( t ) ∈ B for some t ∈ [ 0 , T ]) , D c ( P 0 , P ) ≤ δ where P 0 is a suitable model. P 0 = proxy for P true . P 0 right trade-off between fidelity and tractability. δ is the distributional uncertainty size. Blanchet (Columbia U. and Stanford U.) 28 / 60

A Distributionally Robust Risk Analysis Formulation Our solution: Estimate u T by solving sup P true ( R ( t ) ∈ B for some t ∈ [ 0 , T ]) , D c ( P 0 , P ) ≤ δ where P 0 is a suitable model. P 0 = proxy for P true . P 0 right trade-off between fidelity and tractability. δ is the distributional uncertainty size. D c ( · ) is the distributional uncertainty region. Blanchet (Columbia U. and Stanford U.) 28 / 60

Desirable Elements of Distributionally Robust Formulation Would like D c ( · ) to have wide flexibility (even non-parametric). Blanchet (Columbia U. and Stanford U.) 29 / 60

Desirable Elements of Distributionally Robust Formulation Would like D c ( · ) to have wide flexibility (even non-parametric). Want optimization to be tractable. Blanchet (Columbia U. and Stanford U.) 29 / 60

Desirable Elements of Distributionally Robust Formulation Would like D c ( · ) to have wide flexibility (even non-parametric). Want optimization to be tractable. Want to preserve advantages of using P 0 . Blanchet (Columbia U. and Stanford U.) 29 / 60

Desirable Elements of Distributionally Robust Formulation Would like D c ( · ) to have wide flexibility (even non-parametric). Want optimization to be tractable. Want to preserve advantages of using P 0 . Want a way to estimate δ . Blanchet (Columbia U. and Stanford U.) 29 / 60

Connections to Distributionally Robust Optimization Standard choices based on divergence (such as Kullback-Leibler) - Hansen & Sargent (2016) � � dv �� D ( v || µ ) = E v log . d µ Blanchet (Columbia U. and Stanford U.) 30 / 60

Connections to Distributionally Robust Optimization Standard choices based on divergence (such as Kullback-Leibler) - Hansen & Sargent (2016) � � dv �� D ( v || µ ) = E v log . d µ Robust Optimization: Ben-Tal, El Ghaoui, Nemirovski (2009). Blanchet (Columbia U. and Stanford U.) 30 / 60

Connections to Distributionally Robust Optimization Standard choices based on divergence (such as Kullback-Leibler) - Hansen & Sargent (2016) � � dv �� D ( v || µ ) = E v log . d µ Robust Optimization: Ben-Tal, El Ghaoui, Nemirovski (2009). Big problem: Absolute continuity may typically be violated... Blanchet (Columbia U. and Stanford U.) 30 / 60

Connections to Distributionally Robust Optimization Standard choices based on divergence (such as Kullback-Leibler) - Hansen & Sargent (2016) � � dv �� D ( v || µ ) = E v log . d µ Robust Optimization: Ben-Tal, El Ghaoui, Nemirovski (2009). Big problem: Absolute continuity may typically be violated... Think of using Brownian motion as a proxy model for R ( t ) ... Blanchet (Columbia U. and Stanford U.) 30 / 60

Connections to Distributionally Robust Optimization Standard choices based on divergence (such as Kullback-Leibler) - Hansen & Sargent (2016) � � dv �� D ( v || µ ) = E v log . d µ Robust Optimization: Ben-Tal, El Ghaoui, Nemirovski (2009). Big problem: Absolute continuity may typically be violated... Think of using Brownian motion as a proxy model for R ( t ) ... Optimal transport is a natural option! Blanchet (Columbia U. and Stanford U.) 30 / 60

Application 1: Back to Classical Risk Problem Suppose that c ( x , y ) = d J ( x ( · ) , y ( · )) = Skorokhod J 1 metric. = φ ( · ) bijection { sup inf | x ( t ) − y ( φ ( t )) | , sup | φ ( t ) − t |} . t ∈ [ 0 , 1 ] t ∈ [ 0 , 1 ] Blanchet (Columbia U. and Stanford U.) 31 / 60

Application 1: Back to Classical Risk Problem Suppose that c ( x , y ) = d J ( x ( · ) , y ( · )) = Skorokhod J 1 metric. = φ ( · ) bijection { sup inf | x ( t ) − y ( φ ( t )) | , sup | φ ( t ) − t |} . t ∈ [ 0 , 1 ] t ∈ [ 0 , 1 ] If R ( t ) = b − Z ( t ) , then ruin during time interval [ 0 , 1 ] is B b = { R ( · ) : 0 ≥ t ∈ [ 0 , 1 ] R ( t ) } = { Z ( · ) : b ≤ sup Z ( t ) } . inf t ∈ [ 0 , 1 ] Blanchet (Columbia U. and Stanford U.) 31 / 60

Application 1: Back to Classical Risk Problem Suppose that c ( x , y ) = d J ( x ( · ) , y ( · )) = Skorokhod J 1 metric. = φ ( · ) bijection { sup inf | x ( t ) − y ( φ ( t )) | , sup | φ ( t ) − t |} . t ∈ [ 0 , 1 ] t ∈ [ 0 , 1 ] If R ( t ) = b − Z ( t ) , then ruin during time interval [ 0 , 1 ] is B b = { R ( · ) : 0 ≥ t ∈ [ 0 , 1 ] R ( t ) } = { Z ( · ) : b ≤ sup Z ( t ) } . inf t ∈ [ 0 , 1 ] Let P 0 ( · ) be the Wiener measure want to compute sup P ( Z ∈ B b ) . D c ( P 0 , P ) ≤ δ Blanchet (Columbia U. and Stanford U.) 31 / 60

Application 1: Computing Distance to Bankruptcy So: { c B b ( Z ) ≤ 1 / λ ∗ } = { sup t ∈ [ 0 , 1 ] Z ( t ) ≥ b − 1 / λ ∗ } , and � � Z ( t ) ≥ b − 1 / λ ∗ sup P ( Z ∈ B b ) = P 0 sup . D c ( P 0 , P ) ≤ δ t ∈ [ 0 , 1 ] Blanchet (Columbia U. and Stanford U.) 32 / 60

Application 1: Computing Uncertainty Size Note any coupling π so that π X = P 0 and π Y = P satisfies D c ( P 0 , P ) ≤ E π [ c ( X , Y )] ≈ δ . Blanchet (Columbia U. and Stanford U.) 33 / 60

Application 1: Computing Uncertainty Size Note any coupling π so that π X = P 0 and π Y = P satisfies D c ( P 0 , P ) ≤ E π [ c ( X , Y )] ≈ δ . So use any coupling between evidence and P 0 or expert knowledge. Blanchet (Columbia U. and Stanford U.) 33 / 60

Application 1: Computing Uncertainty Size Note any coupling π so that π X = P 0 and π Y = P satisfies D c ( P 0 , P ) ≤ E π [ c ( X , Y )] ≈ δ . So use any coupling between evidence and P 0 or expert knowledge. We discuss choosing δ non-parametrically momentarily. Blanchet (Columbia U. and Stanford U.) 33 / 60

Application 1: Illustration of Coupling N ( t ) Given arrivals and claim sizes let Z ( t ) = m − 1 / 2 ∑ k = 1 ( X k − m 1 ) 2 Blanchet (Columbia U. and Stanford U.) 34 / 60